Backward Euler method

In numerical analysis and scientific computing, the backward Euler method is one of the most basic numerical methods for the solution of ordinary differential equations. It is similar to the standard Euler method, but it differs in that it is an implicit method, so each step defines the new value through an equation rather than computing it directly from the old one. The lead states that the backward Euler method has error of order one in time. Its place in the field is as a first, simple member of the family of implicit time-stepping schemes, and it is usually contrasted with the explicit forward form that shares its name.